Annals o Pu e and Applied Logic 175 (2024) 103342
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Annals o Pu e and Applied Logic
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A logico-geome ic compa ison o cohe ence o non-addi i e
unce ain y measu es
Es he Anna Co si a,˚, Tommaso Flaminio b, Hykel Hosni a
aLUCI G oup, Depa men o Philosophy, Uni e si y o Milan, I aly
bA ificial In elligence Resea ch Ins i u e, IIIA – Spanish Na ional Resea ch Council, CSIC, Spain
a i c l e i n o a b s a c
A icle his o y:
A ailable online 17 July 2023
MSC:
03B48
60A05
68T37
60A10
Keywo ds:
Cohe ence
P obabili y logic
Belie unc ions
Lowe p obabili y
We in es iga e he no ion o cohe ence o (non-)addi i e unce ain y measu es om
a logico-geome ic poin o iew. Ou main esul is o he effec ha dis inc
c i e ia o cohe ence a e no always ma ched by axioma ically dis inc measu es
o unce ain y. In addi ion we in oduce a me alogic wi hin which his kind o esul
can be cap u ed o mally.
© 2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle
unde he CC BY license (h p://c ea i ecommons .o g /licenses /by /4 .0/).
1. In oduc ion and mo i a ion
The in es iga ion epo ed in his pape has i s o igins is [24,21,9]and iden ifies he common oo o
logic and p obabili y in he no ion o cohe ence. Ou key esea ch ques ion is whe he dis inc c i e ia o
cohe ence a e always ma ched by axioma ically dis inc measu es o unce ain y. Mo eo e we ask whe he
his dis inc ion, when a ailable, can be exp essed wi hin a sui able me alogic.
In pa icula , we s udy he no ion o cohe ence o se e al unce ain y measu es by geome ic means. By
le e aging on his geome ic ep esen a ion we show ha cohe ence is no sufficien o dis inguish books on
non i ial se s o e en s ha a e ex endible o lowe p obabili ies and belie unc ions. The same geome ic
ep esen a ion will be hen used o a pu ely logical analysis o cohe ence ha will be done by Riesz infini e-
alued logic.
Since cohe ence ea u es a (pe haps unique) combina ion o logico-ma hema ical, ounda ional and p ac-
ical in e es , we begin by helping he eade o app aise he wide landscape which he p esen wo k helps
*Co esponding au ho .
E-mail add esses: [email p o ec ed] (E.A. Co si), [email p o ec ed] (T. Flaminio), hyk[email p o ec ed] (H. Hosni).
h ps://doi.o g/10.1016/j.apal.2023.103342
0168-0072/© 2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://
c ea i ecommons .o g /licenses /by /4 .0/).
2E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
sys ema ising. This will also se e he pu poses o in oducing some o he e minology and no a ion ha
will be used h oughou he pape .
Fo he ounde s o ma hema ical logic, Boole and De Mo gan, logic and p obabili y belonged on equal
e ms o he ma hema ical analysis o sound in e ence. In he in e ening wo cen u ies hough, he fields
ha e g own la gely independen ly, as is appa en om ex book p esen a ions. And ye he e is much o be
unde s ood abou he ounda ions and applica ions o easoning unde unce ain y by looking e y closely
a how logic and p obabili y concu in o ming a cons ella ion o me hods, models and axioma isa ions
[33,37,28].
Logic plays a wo old ole in he ounda ions and applica ions o unce ain easoning. Syn ac ically,
i p o ides an unambiguous defini ion o he objec s o an agen ’s easoning, i.e. he algeb a o e en s.
Seman ically, i p o ides he mechanism necessa y o he agen o quan i y he unce ain y which is no
esol ed by he in o ma ion hey possess. In he mos amilia case o p obabili y, wo Boolean algeb as a e
a wo k (see Subsec ion 2.1 below o basic e minology): A, which o malises e en s, and 2, which o malises
he alues ha indica o unc ions ake. Homomo phisms om A o 2co espond o classical e alua ions
o o mulas, i.e. in ag eemen wi h Ta skian seman ics. E en s which do no ge a bina y u h- alue a e
he objec s o he dis ibu ion o a p obabili y mass unc ion, which is done acco ding o c i e ia exceeding
he seman ics o classical logic.
Agains his backg ound he classical p oblem o cohe ence can be pu as ollows:
Gi en
1. An idealised agen who is equi ed o quan i y hei unce ain y conce ning a (fini e) subse o A,
Ψ“ a1,a
2,...,a
nu.
2. Unce ain y is esol ed by Ta skian seman ics using e alua ions as specified abo e.
Wan he condi ions unde which he assignmen
β:ΨQaiÞÑ βiP 0,1s
can be said o be cohe en .
In his con ex , he e m “cohe ence” has been popula ised by B uno de Fine i [11,12], who a gued pe sua-
si ely ha he desi ed condi ions should be ep esen ed by no malisa ion and fini e addi i i y (see below),
bu as we shall ecall, de Fine i’s wo d was a om being he las one on he subjec . No e ha cohe ence
bea s he ollowing deside a a. We equi e β o
1. ep esen he in o ma ion (i any) a ailable o he agen ;
2. pin down he app op ia e quan ifica ion o he unce ain y which is no esol ed by in o ma ion.
Fo p esen pu poses, in o ma ion esol es unce ain y o he ex en ha he (classical, p oposi ional)
sen ences which ep esen i decide o he sen ences o in e es . So o ins ance, he in o ma ion ep esen ed
by φ esol es (classically) any unce ain y abou φ _ψ. No e ha he con e se does no hold.
Thus he p oblem o cohe ence boils down o iden i ying he condi ions unde which βquan ifies un-
ce ain y “app op ia ely”. This is an ex a-logical equi emen , which ypically depends on he pu pose o
which we a e conside ing a model o unce ain y quan ifica ion in he fi s place. Following a adi ion
which has played a c ucial ole in he de elopmen o he heo y o p obabili y, we es ic ou a en ion
o si ua ions in which he agen ’s decision-making is spel -ou in e ms o (highly abs ac and idealised)
be ing beha iou . Depending on he na u e o he be ing p oblem, ce ain deside a a become pa icula ly
compelling, as discussed in Sec ion 2, and summa ised in Table 1.
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 3
Table 1
A summa y o he ela ion be ween he key axioma ic
p ope ies e sus he cha ac e is ics o he decision
p oblems and he co esponding unce ain y esolu ions.
The ele an defini ions a e p o ided in Sec ion 2.
Key axiom Be ing Unce ain y esolu ion
Addi i i y 2-sided Classical
8-mono onici y 2-sided Pa ial
Supe -addi i i y 1-sided Classical
A med wi h his li le e minology and no a ion, we can p oceed o loca e mo e p ecisely he p esen
con ibu ion ela i e o he wide a ea o p obabili y logic. The p oblem o cohe ence as we ecalled i ,
was ini ially mo i a ed by ounda ional conce ns which we e absen om he coe al in es iga ions on he
ma hema ical p oblem o iden i ying he condi ions o a Boolean algeb a o ca y a fini ely addi i e measu e.
This line o in es iga ion is oo ed in on Neumann’s ea ly wo k on σ-algeb as da ing back o he la e 1930’s,
and goes h ough a conjec u e o Ho n’s and Ta ski’s, i s e u a ion due o Gai man, and culmina es wi h
he 1959 ep esen a ion o Kelley’s – see [35] o a e se accoun .
In addi ion he measu e heo e ic ques ions, which does no cons i u e he ocus o ou wo k, he ea ly
1980’s wi nessed an explosion o in e es in p obabilis ic ex ensions o logical in e ence owing o he p omises
o expe sys ems (see [39] o a comp ehensi e his o ical o e iew). This b ough he p oblem o combin-
ing logic and p obabili y o he a en ion o he AI communi y see, e.g. [42,20,26,28]. Wi h a peculia
“applica ions- ounda ions eedback”, anumbe o c i icisms o he p obabilis ic ep esen a ion o unce -
ain y became commonplace wi hin AI. Among hem o he inabili y o (fini ely) addi i e measu es o
unce ain y o ep esen wo impo an aspec s o in o ma ion p ocessing sys ems, namely agueness and
pa ial igno ance.
The o me conce n a ises because no all e en s o in e es o an (idealised, a ificially in elligen ) agen
ha e a bina y ealisa ion. To he con a y, many p ope ies a e bes hough o as g aded, om a pe son’s
age, o he business o a oad junc ion. Wi h his mo i a ion, [58,59] pu o wa d an ex ension o classical
logic, whe e u h- alues in 0, 1sa e in e p e ed as “deg ees o u h” o uzzy sen ences. Since uzziness
is a seman ic (i.e. logical) p ope y o he unce ain y esolu ion is dis inc om a measu e quan i ying
unce ain y. Hence i is ma hema ically meaning ul and p ac ically use ul o unde s and how say, p obabili y
should be assigned o non-bina y e en s. This leads o a enewed in e es in he ea ly 20 h cen u y app oaches
o many- alued logics, [29]and in pa icula he Łukasiewicz eal- alued logic [41]. By he beginning o his
cen u y i became clea ha he as li e a u e ex ending classical logics could be used in p obabilis ic
unce ain y esolu ion. Fo ou p esen pu poses, apa icula ly impo an con ibu ion in his espec is
[44]. In i Pa is shows ha one o he key me hods o jus i ying he quan ifica ion o unce ain y by means
o p obabili y – he Du ch Book me hod o be discussed a leng h below – does no necessi a e Ta skian
unce ain y esolu ion. The Riesz consequence ela ion which plays a cen al ole in Sec ion 4below, sums up
much o he unexpec ed in e ac ion be ween many- alued logics and p obabili y which has been mo i a ed
by Pa is’s no e.
As o pa ial igno ance, he conce n a ises because p obabili y may o ce agen s o go unwa an edly
beyond he unce ain y esol ed by Ta skian seman ics. We illus a e wi h a classic p oblem popula ised by
Ellsbe g [18].
Example 1.1. Conside an u n wi h ed, blue and g een balls. Suppose ψ1s ands o “ he ball is ed”, ψ2
s ands o “ he ball is blue and ψ3s ands o “ he ball is g een”. Suppose u he ha he agen knows ha
he p opo ion o he ed balls in he u n is 1/3. Rep esen ing his in o ma ion p obabilis ically, leads o he
s aigh o wa d quan ifica ion o he agen ’s unce ain y in he ele an e en , i.e. Ppψ1q “1{3. Obse e
now ha he in o ma ion a ailable does no esol e “enough” unce ain y o allow he agen o come up wi h
equally s aigh o wa d quan ifica ions o Ppψ2) and Ppψ3q. Any alue in 0, 2{3swill be consis en wi h
4E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
he in o ma ion a ailable. In he absence o any u he in o ma ion, he p obabilis ic se ing would lead o
se ing bo h Ppψ2qand Ppψ3q o 1{3. The p oblem wi h ha is ha 1{3 ails o ep esen he in o ma ion
o he effec ha he in o ma ion a ailable does no suppo 1{3any mo e han any alue in 0, 2{3s.
A popula eac ion o his kind o p oblem has been o weaken he addi i i y o p obabili y unc ions.
Among he many p oposals in his di ec ion, wo ha e p o ed o be pa icula ly ui ul. Following ea lie
s a is ical wo k o Demps e , Sha e [49] sough o cap u e he unce ain y esolu ion p o ided by classi-
cal logic as de e mining he e iden ial suppo o a supe -addi i e measu e, which has become known as
Demps e -Sha e belie unc ion ( o be defined below). The second non-addi i e app oach o measu ing he
igno ance a ising om pa ial unce ain y esolu ion, consis s in aking (con ex) se s o p obabili y unc-
ions – he ones which a e consis en wi h he in o ma ion a ailable. This esul s in so-called c edal se s [38]
which lead na u ally o defining lowe (and uppe ) p obabili ies o he e en s o in e es . This app oach,
which is oo ed in he in es iga ion o Inne and Ou e measu es [30], has been championed by Walley [54].
Like belie unc ions, lowe p obabili ies ( o be defined below) a e also supe -addi i e.
In ligh o his ough and incomple e ske ch o he landscape, i is no pa icula ly su p ising ha a
g ea a ie y o app oaches o logic-based unce ain easoning ha e been pu o wa d o e he pas ew
decades, each ollowing i s own peculia blend o ma hema ical, ounda ional and p ac ical mo i a ion. As
a esul i may no be always ob ious which amewo k o measu e o unce ain y is bes sui ed o which
kind o p oblem. The main aim o he p esen pape is o pu o wa d logico-geome ic ools ha can
con ibu e significan ly o ob aining a unified pic u e. Fa om being only a p oblem o applica ions, ying
he p ope ies o unce ain y measu es wi h he kind o si ua ions in which hey a e p ac ically use ul has
a dis inc ounda ional impo ance.
The emainde o pape is o ganised as ollows. Sec ion 2 e iews he measu es o unce ain y o p esen
in e es (Subsec ion 2.2) and hei associa ed no ions o cohe ence (Subsec ions 2.3–2.5). Whils he e-
sul s o his Sec ion a e no no el, he way we p esen hem can be o independen in e es , in addi ion
o an icipa ing, h ough nume ical examples, some key geome ic insigh s ha will be used in ou main
esul s. Sec ion 3 o ms he co e o ou pape . In Subsec ion 3.1 we ocus on he axioma ic compa ison o
lowe p obabili ies and belie unc ion. Using he geome ic amewo k in oduced in Subsec ion 3.2, Subsec-
ion 3.3 p esen s he main esul o his pape , Theo em 3.10. In i we iden i y a he mild condi ions unde
which belie unc ions and lowe p obabili y a e cohe ence-wise indis inguishable despi e being axioma i-
cally dis inc . Sec ion 4showcases a me a heo ic ole o many- alued logics in easoning abou cohe en
measu es o unce ain y. In i we define he Riesz consequence ela ion and show in Theo em 4.10 how i
can ep esen o mally he geome ic cohe ence-wise compa isons pu o wa d in Subsec ion 3.3. As we shall
obse e in he concluding Sec ion 5, his me alogical se ing may p o e use ul in he abs ac in es iga ion
o cohe ence-based unce ain y measu es.
2. P elimina ies
This Sec ion begins by ecalling he basic defini ions and esul s on Boolean algeb as (Subsec ion 2.1)
and o h ee key measu es o unce ain y and hei associa ed dual measu es (Subsec ion 2.2). In i , we shall
highligh he p ope ies which will be o pa icula in e es o ou esul s. Then we e iew he o iginal se up
due o de Fine i (Sec ion 2.3) and i s ex ension o pa ly esol ing unce ain y by Jaff ay in Sec ion 2.4
and o imp ecise p obabili ies in Sec ion 2.5. No e ha his Sec ion ecalls ma e ial selec ed wi h he goal
o making he pape essen ially sel -con ained and does no a emp o do jus ice o he inc edibly ich
ele an li e a u e, o which excellen su eys a e a ailable, including [43,33,37,28,52]. Again in he in e es
o b e i y, some basic logico-algeb aic no ions a e aken o g an ed. We e e , wi h apologies, eade s who
need o fill ou he gaps o [7,27,47].
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 5
2.1. Fini e Boolean algeb as and hei a oms
Fini e Boolean algeb as a e he algeb aic amewo k o his pape ; hence i s logical se ing is ha o
classical p oposi ional logic, CPL. He e, we will b iefly ecap on some needed no ions and basic esul s
abou Boolean algeb as and CPL, o a mo e exhaus i e in oduc ion abou his subjec we in i e he
eade o consul [47], and [6,32,31].
Gi en a coun able (fini e o infini e) se Vo p oposi ional a iables, he CPL language LpVq(o simply L
when Vwill be clea by he con ex ) is he smalles se con aining Vand closed unde he usual connec i es
^, _, , K, and Jo ype p2, 2, 1, 0, 0q. Along his pape we will use he no a ion ϕ, ψ, e c (wi h possible
subsc ip ) o o mulas. Fu he , we shall adop he ollowing abb e ia ions:
ϕ Ñψ“ϕ _ψ, ϕ Øψ“pϕ Ñψq ^pψÑϕq.
We shall deno e by $CPL he p o abili y ela ion o CPL, in pa icula we will w i e $CPL ϕ o deno e
ha ϕis a heo em.
A logical alua ion (o simply a alua ion) o Lis a map om V o he domain 0, 1uo he Boolean
algeb a 2, which uniquely ex ends o a unc ion, ha we deno e by he same symbol , om L o 0, 1uin
acco dance wi h he usual Boolean u h unc ions, i.e. pϕ ^ψq “min pϕq, pψqu, pKq “0, pϕq “
1 ´ pϕq, e c. We shall deno e by Ω he se o all alua ions o L. Fo a gi en o mula ϕand a gi en alua ion
PΩ, we will w i e |ù ϕwhene e pϕq “1.
We will b oadly adop , analogously o he abo e ecalled logical ame, he signa u e p^, _, , K, Jq
o ype p2, 2, 1, 0, 0q o he algeb aic language upon which Boolean algeb as a e defined. Thus, he same
con en ions and abb e ia ions o Lcan be adop ed also in he algeb aic se ing. Fu he , in e e y Boolean
algeb a A “pA, ^, _, , K, Jq we shall w i e a ďb, whene e a Ñb “J. The ela ion ďis indeed he
la ice-o de in A. Thus, a ďbiff a ^b “aiff a _b “b.
Along his pape , in o de o dis inguish an algeb a om i s uni e se, we will deno e he o me by A,
Be c, and he la e by A, Be c, espec i ely.
Recall ha a map h :A ÑBbe ween Boolean algeb as is a homomo phism i hcommu es wi h he
ope a ions o hei language, ha is, hpJAq “J
B, hpAaq “
Bhpaq, hpa ^Abq “hpaq ^Bhpbqe c
(no ice ha we adop subsc ip s o dis inguish he ope a ions o A om hose o B). Bijec i e (o 1-1)
homomo phisms a e called isomo phisms and i he e is a isomo phism be ween Aand B, hey a e said o
be isomo phic (and we w i e A –B).
Defini ion 2.1. An elemen αo a Boolean algeb a Ais said o be an a om o Ai αąKand o any o he
elemen b PAsuch ha αěb ěK, ei he α“bo b “K.
E e y fini e algeb a has a oms ha will be deno ed by α, β, γe c.
2.2. Unce ain y measu es
As we al eady s a ed be o e, in his pape we only conside fini e, and hence a omic, Boolean algeb as.
Those a e he domains o he unce ain y measu es we deal wi h.
Defini ion 2.2 (P obabili y unc ions). A p obabili y unc ion on an algeb a Ais a 0, 1s- alued map P
sa is ying:
(P1) PpJq “1, PpKq “0; (No malisa ion)
(P2) Ppa _bq “Ppaq `Ppbq, i a ^b “K. (Fini e addi i i y)

6E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
The se o all p obabili y unc ions on Ais deno ed by PA. We omi he supe sc ip when he algeb a is
clea om he con ex .
Rema k 2.3. I is cus oma y in measu e- heo e ic app oaches o cas Defini ion 2.2 wi hin a p obabili y
space, i.e. a coun able se o elemen a y ou comes Ωwi h an algeb a Aon i , ypically a field o se s. Then
one defines a measu e μon Aimposing no malisa ion on Ωand coun able addi i i y, which gene alises (P2)
abo e o coun able unions o (coun ably many incompa ible) e en s. Fo ou pu poses i is sufficien o limi
ou sel es o fini e addi i i y, no ing ha s anda d esul s, which s a wi h de Fine i’s own, show ha
he e is no loss o gene ali y in doing his, see [3] o a comp ehensi e o e iew. In addi ion, fini e addi i i y
p o ides he na u al se ing o p obabili y logic, see Chap e 3 o [45].
Defini ion 2.4 (Belie unc ions [49]). A belie unc ion on an algeb a Ais a 0, 1s- alued map Bel sa is ying:
(B1) BelpJq “1, BelpKq “0;
(B2) Bel˜n
ł
i“1
ai¸ě
n
ÿ
i“1ÿ
JĎ 1,...,nu:|J|“iu
p´1qi`1Bel˜ľ
jPJ
aj¸, o n PN.
Rema k 2.5. No e ha (B2) does no ake he o m o a s aigh o wa d gene alisa ion o (P2):
Belpθ_φqěBelpθq`Belpφq,i θ^φ“K.(Fini e supe -addi i i y)
Belie unc ions a e indeed hose supe -addi i e se unc ions which a e comple ely addi i e, o mono one,
as i is some imes said.
In he fini e se ing, belie unc ions on Boolean algeb as can be cha ac e ized in e ms o he associa ed
mass unc ions as ollows. Le Abe any fini e Boolean algeb a wi h a oms α1, ..., α . Amass unc ion is a
map m ha assigns o each subse Xo a oms (called ocal elemen s) a eal numbe such ha mpHq “0
and řXmpXq “1. Gi en a mass unc ion m, he map
Belpaq“ ÿ
XĎ αi|αiďau
mpXq(1)
is a belie unc ion and e e y belie unc ion on Aa ises om (1).
Example 2.6. Conside he fini e Boolean algeb a Awi h wo a oms α1and α2and he ollowing mass
assignmen :
mp α1uq “ 0.2
mp α2uq “ 0.4
mp α1,α
2uq “ 0.4
mpHq “ 0.
Since mpHq “0and řXĎAmpXq “mp α1, α2uq `mp α1uq `mp α2uq `mpHq “1, i is he case ha
mis a mass unc ion. Mo eo e , since řyĎ α1umpyq “0.2and řyĎ α2umpyq “0.4, he map β:α1ÞÑ
0.2, α2ÞÑ 0.4is a belie unc ion.
An elemen ϕo a Boolean algeb a Ais said o be co e ed m imes by a mul ise a1, ..., anuu o elemen s
o Ai e e y homomo phism o A o 0, 1u ha maps ϕ o 1, also maps o 1a leas mp oposi ions om
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 7
a1, ..., anas well. An pm, kq-co e o pϕ, Jq is a mul ise a1, ..., anuu ha co e s Jk imes and co e s ϕ
n `k imes.
Defini ion 2.7 (Lowe p obabili y unc ions [51]). A lowe p obabili y on an algeb a Ais a mono one 0, 1s-
alued map Psa is ying:
(L1) PpJq “1, PpKq “0;
(L2) Fo all na u al numbe s n, m, kand all a1, ..., an, i a1, ..., anuu is an pm, kq-co e o pϕ, Jq, hen
k`mPpϕq ě
n
ÿ
i“1
Ppaiq.
Al hough his defini ion does no make he name lowe p obabili ies pa icula ly ob ious, [1, Theo em 1]
pu s o wa d he ollowing enligh ening cha ac e isa ion, an icipa ed by [51]. Le P:A Ñ 0, 1sbe a
lowe p obabili y and deno e wi h MpPq he se o p obabili y unc ions which bound P om abo e, i.e.
MpPq “ PPP|Ppaq ďPpaq, @a PAu. Then, o all a PA,
Ppaq“ min
PPMpPqPpaq.(2)
Example 2.8. Le us conside he fini e Boolean algeb a Awi h wo a oms α1and α2and he assignmen
β:K ÞÑ 0, α1ÞÑ 0.2, α2ÞÑ 0.4, J ÞÑ 1. βis a lowe p obabili y. To see his no e ha Mpβq “ PPP|
0.2 ďPpα1q, 0.4 ďPpα1qu. In pa icula , he p obabili y unc ions P1and P2 ha belong o Mpβqand
ha gene a e βa e P1:α1ÞÑ 0.2, α2ÞÑ 0.8and P2:α1ÞÑ 0.6, α2ÞÑ 0.4, espec i ely.
Rema k 2.9. Belie unc ions and lowe p obabili ies cons i u e he bes -known axioma ic gene alisa ions o
p obabili y unc ions. As i is appa en om he defini ions abo e, p obabili y unc ions a e he addi i e
special case o bo h belie unc ions and lowe p obabili ies. Mo eo e , belie unc ions and lowe p obabili ies
can be ela ed h ough p obabili y unc ions. Howe e , he ques ion o in e p e ing his ela ion in e ms o
cohe ence, u ns ou o be ema kably difficul , as he main esul s o ou pape illus a e in Sec ion 3.
We end his subsec ion by ecalling he defini ion o wo o dinal measu es o unce ain y which s and in
duali y ela ion o one ano he , and which will play an impo an ole in wha ollows.
Defini ion 2.10 (No malized necessi y measu es). A no malized necessi y measu e on an algeb a Ais a 0, 1s-
alued map Nsa is ying:
(N1) NpJq “1, NpKq “0;
(N2) Npa1^a2q “min Npa1q, Npa2qu.
To each necessi y measu e on a Boolean algeb a Ais associa ed a dual possibili y measu e, usually deno ed
by Πand defined on Aby le ing
Πpaq“1´Npaq,(3)
o all a PA– see [17,33] o de ails.
Each possibili y measu e Πon a fini e Boolean algeb a Agi es a no malized possibili y dis ibu ion π
once es ic ed o he a oms α1, ..., α o A. In his con ex , no maliza ion means ha πpαiq “1, o
a leas one αi. Con e sely, each no malized possibili y dis ibu ion πon he αi’s uniquely de e mines a
possibili y Πand a necessi y measu e Nby he ollowing s ipula ions:
8E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
Fig. 1. The unce ain y measu es conside ed so a a anged by gene ali y (solid a ows) and hei dual companions (dashed a ows).
Πpaq“
ł
j“1
πpαjq^apαjqand Npaq“
ľ
j“1
p1´πpα qq_apαjq,(4)
whe e apαjqs ands o 1i αjďaand 0o he wise.
Rema k 2.11 (Dual unce ain y measu es). In he same way as possibili y measu es can be defined by duali y
in he sense o equa ion (3), dual companions (also known as conjuga e measu es) can be defined also o he
o he unce ain y measu es we conside ed so a . Mo e p ecisely, while p obabili y unc ions and sel -dual
in he sense ha e e y p obabili y Pon a Boolean algeb a Asa isfies Ppaq “1 ´Ppaq o all a PA, he
ollowing cases a ise o non-addi i e measu es:
•Fo e e y belie unc ion Bel on A, he map Pl :A Ñ 0, 1ssuch ha o all a PA, Plpaq “1 ´Belpaq
is called plausibili y unc ion.
•The dual companion o a lowe p obabili y P:A Ñ 0, 1s, defined as Ppaq “1 ´Ppaqis known in
he li e a u e as an uppe p obabili y unc ion.
Needless o say ha , in gene al, Bel and Pl, and Pand Pdiffe on a Boolean algeb a. Indeed, i on a
Boolean algeb a A, one has ha o all a PA, Belpaq “Plpaq(o Ppaq “Ppaq), hen Bel (P espec i ely)
is a p obabili y unc ion. Mo e de ails can be ound in [33].
The diag am depic ed in Fig. 1sums up he mu ual ela ions be ween he unce ain y measu es and hei
duals desc ibed in his Subsec ion.
2.3. Two-sided be ing wi h ully esol able unce ain y
As an icipa ed in Sec ion 1, he main ocus o his pape is on cohe ence. As his no ion has been he
adema k o he ounda ional poin o iew championed by B uno de Fine i, we begin by ecalling he
amewo k he laid down in his seminal [11].
Suppose ha a1, ..., ana e elemen s o a fini e Boolean algeb a A, which a e in e p e ed as he e en s
o in e es o a bookmake B. Suppose ha his in e es ma e ialises wi h he publica ion o a book β:a1ÞÑ
β1, ..., anÞÑ βnwhe e βiP 0, 1s o i “1, ..., n. This assignmen is made by Bunde a numbe o
cons ain s. The mos impo an o which o ces B o le a gamble Gchoose eal- alued s akes σ1, ..., σn
o each aiin he book. Hence, o i “1, ..., n, Gpays σiβi o Bin e u n o σi paiq. Thus, G’s payoff is
řn
i“1σip paiq ´βiqwhe eas B’s payoff is řn
i“1σipβi´ paiqq.
Rema k 2.12. Th oughou is a (Boolean algeb a) homomo phism om A o 0, 1uwhich ag ees, ha is,
wi h he classical p oposi ional seman ics. Hence unce ain y is esol ed classically – and we use he no a ion
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 9
as a eminde ha Boolean algeb a homomo phisms play he ole o classical p oposi ional alua ions ( ecall
Subsec ion 2.1). We e e ed o his si ua ion as he “classical p oblem o cohe ence” in Sec ion 1.
This se -up is sufficien o de Fine i o pu o wa d a defini ion which has had a p o ound impac on
he ounda ions o p obabili y since he second hal o he nine een h cen u y.
Defini ion 2.13 (Cohe ence). βis cohe en i he e is no a1, ..., anPAand σ1, ..., σnin Rsuch ha o
e e y ,
n
ÿ
i“1
σipβi´ paiqq ă 0.(5)
Equi alen ly, βis cohe en i o e e y a1, ..., anPAand σ1, ..., σnPR, he e is a such ha ,
n
ÿ
i“1
σipβi´ paiqq ě 0.(6)
In o he wo ds, he book published by Bis cohe en i he e is no choice o e en s and o (possibly nega i e)
s akes which Gcan make, exposing B o a su e loss.
As shown in [11,12], cohe ence is necessa y and sufficien o he exis ence o a fini ely addi i e measu e
P ha ex ends he book βo e A, i.e. a p obabili y unc ion Psuch ha o i “1, ..., n, Ppaiq “βi.
De Fine i sough o use his esul o jus i y he ep esen a ion o “ a ional deg ees o belie ” by means o
p obabili ies, a line o easoning which has become known as he Du ch Book a gumen . The e a e many
mo e de ails o his a gumen , which a e imma e ial o ou p esen pu poses (we e e in e es ed eade s
o [23] o a logical p esen a ion consonan o he p esen se ing). One key poin hough, is ha de Fine i
concoc ed his a gumen in such a way ha Ba oids su e loss exac ly by publishing a ai p ice book, i.e.
one ca ying null expec a ion o ei he gain o loss.
Fai p ices a e among he e y fi s de ices which ha e been used o gi e meaning o p obabili y. They
eflec an idealised and abs ac si ua ion in which e e y aspec o he book is pe ec ly known o B, excep
o cou se, which e en s will e en ually ob ain. As ecalled in Sec ion 1se e al au ho s, s a ing wi h he
ea ly con ibu ions by F ank Knigh and John M. Keynes, ha e aken issue wi h he una ainable demands
o such idealisa ions. O e he pas cen u y, a ich landscape o gene alisa ions o p obabili y measu es has
a isen in esponse o such ounda ional as well as p ac ical conce ns, wi h a pa icula ly in e es ing line o
esea ch aking place wi hin AI. As ecalled abo e, he mo i a ions, as well as he ma hema ical app oaches
pu sued, di e ge significan ly and ye he elaxa ion o addi i i y is one impo an ea u e ha wide a ay
o o malisms ha e in common. Since ou main ocus in Sec ion 3will be on he ela ion be ween lowe
p obabili ies and belie unc ions – wo p ominen non-addi i e measu es – we now e iew how hose a ise
na u ally by sui ably elaxing some condi ions in de Fine i’s a gumen .
2.4. Two-sided be ing wi h pa ly esol able unce ain y
The fi s gene alisa ion we conside is due o J-Y Jaff ay who in [34], cha ac e ised cohe en deg ees o
belie unde pa ly esol ed unce ain y. This led o a ep esen a ion o belie unc ions which he linked o
he hen-eme ging field o decision heo y unde ambigui y (see [53] o a ho ough con ex ualisa ion).
In Jaff ay’s se ing, cohe ence is defined essen ially as in Defini ion 2.13 abo e, excep ha he unde lying
unce ain y esolu ion mechanism is no longe p o ided by classical logic. Ins ead o homomo phisms ,
unce ain y is esol ed by unc ions Capaiqdefined as ollows:
16 E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
in he coming o age o he Du ch Book me hod, as he i le says. In his Subsec ion, we lay down he
geome ic ools, and ele an no a ion/ e minology, which will be used in he es o he pape . We cas in
his amewo k he well-known ex ension esul s yielded by geome ically cohe en assignmen s, collec ed
in Theo em 3.4.
Le Ψ “ a1, ..., anube a fini e se o e en s (i.e., elemen s o a fini e Boolean algeb a A). Le us
deno e by V“ 1, ..., u he fini e se o all possible homomo phisms o A o he boolean chain on he
wo-elemen se 0, 1u. Fo e e y j“1, ..., , call ej he bina y ec o
ej“p jpa1q,...,
jpanqq P 0,1un.(14)
Gi en his basic cons uc ion, and using an app oach simila o Pa is’s we can cha ac e ize in geome ic e ms
he ex endabili y p oblem o books on Ψ o fini ely addi i e p obabili y measu es, no malized necessi y
measu es and belie unc ions. The addi ional no ions we need a e he Euclidean closed con ex hull copXq
o a subse XĎR (which educes o copXqin case Xis fini e) and he less common opical con ex hull
co^,`pXqo X(see [13]).
Defini ion 3.3. Le x1, ..., x P 0, 1sn. The opical hull o he xj’s is he subse co^,`px1, ..., x qo all
poin s yo 0, 1sn o which he e exis pa ame e s λ1, ..., λ P 0, 1ssuch ha Ź
j“1λj“0and
y“
ľ
j“1
λj`xj.
The symbol ^s ands o he minimum and ` o he o dina y addi ion in he opical semi ing pR, ^, `q.
Gi en λ P 0, 1sand xP 0, 1sn, λ `x“pλ `x1, ..., λ `xnqand he Źope a o is defined componen -wise.
Fo e1, ..., e being defined as abo e om he o mulas ai’s in Ψ, le us conside he ollowing se s:
1. PΨ“cope1, ..., e q;
2. NΨ“co^,`pe1, ..., e q;
3. BΨ“copNΨq, whe e, in his case, being NΨusually uncoun able, co deno es he opological closu e o
he Euclidean con ex hull co.
Theo em 3.4 (Geome ic ex ension esul s [12,22,25]). Le Ψ “ a1, ..., anube a fini e se o e en s. Then
a book β:Ψ Ñ 0, 1sex ends o a
#P obabili y measu e
Necessi y measu e
Belie unc ion +i and only i pβpa1q,...,βpanqq P #PΨ
NΨ
BΨ+.
In gene al, PΨand NΨa e bo h s ic ly included in BΨ(i.e., PΨĂBΨand NΨĂBΨ) and his is
expec ed because belie unc ions a e s ic ly mo e gene al han bo h p obabili ies and no malized necessi y
measu es. In he p esen wo k, we in es iga e, ia cohe ence, whe he i is possible o dis inguish unce ain y
heo ies when we conside he mo e gene al se ing o lowe p obabili ies. In pa icula , we s udy i cohe ence
is sufficien ly obus o dis inguish lowe p obabili ies om belie unc ions.
Le us no ice ha , o e e y subse o e en s Ψas abo e, he se s PΨ, NΨand BΨa e polyhed a o
0, 1sn. Mo e p ecisely, PΨand BΨa e poly opes, i.e. con ex polyhed a in he usual Euclidean sense, while
NΨis con ex in he opical sense specified in Defini ion 3.3 abo e, bu i is no con ex in he s anda d
Euclidean model o opical geome y. Thus, in ha s anda d model, NΨis ep esen ed by a polyhed on.

E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 17
As o lowe p obabili ies, he si ua ion is simila bu no ully unde s ood. Deno e by LΨ he se o all
books on Ψ ĎA ha ex end o a lowe p obabili y on he Boolean algeb a A. Al hough LΨis known o be
a poly ope [46], a cha ac e iza ion o i s ex emal poin s is no ully unde s ood. In [10](see in pa icula
§9 o he same pape ) he au ho s make his poin pa icula ly clea . Howe e , o he sake o he p esen
pape , and in pa icula o he esul s o his Sec ion and o Sec ion 4, such a ull desc ip ion is no needed.
The nex p oposi ion makes clea a fi s ela ion be ween LΨand PΨ.
P oposi ion 3.5. Le Abe a fini e Boolean algeb a and Ψ “ a1, ..., anu ĎA. Abook βon Ψbelongs o LΨ
i and only i he e a e β1, ..., βnPPΨsuch ha , o all aiPΨ, βpaiq “min βjpaiq |j“1, ..., nu.
P oo . The igh - o-le di ec ion is i ial. Le us hence assume ha βex ends o a lowe p obabili y P.
Le MpPq “ P|Ppaq ěPpaq, @a PAuas in Sec ion 2.2 and hen, o all aiPΨ,
Ppaiq“min Ppaiq|PPMpPqu.
Fo all PPMpPq, call βP he (necessa ily cohe en ) book on Ψ ob ained om Pby es ic ion. Then,
ob iously,
βpaiq“min βPpaiq|PPMpPqu.
Finally, since Ψis fini e, o e e y aiPΨfix a book βPpiqamong he βP’s such ha
βPpiqpaiq“βpaiq“min βPpaiq|PPMpPqu.
Fo e e y i, βPpiqexis s. Then he claim ollows since βpaiq “min βPpaiq |P“Ppiqu. In o he wo ds
β“min βPp1q, ..., βPpnqu.l
The ela ion be ween belie unc ions and lowe p obabili ies desc ibed in Sec ion 3.1 is ai h ully cap u ed
wi hin ou geome ic amewo k.
P oposi ion 3.6. Le Abe a fini e Boolean algeb a and Ψ “ a1, ..., anu ĎA, hen
BΨĎLΨ.
P oo . To p o e ou claim, i is sufficien o show ha all he e ices o BΨbelong o LΨ. The e ices o
BΨa e ei he e1, ..., e o ec o s eco e ed by aking he componen -wise minimum be ween any subse
o e1, ..., e . The e o e, by P oposi ion 3.5, all he e ices o BΨa e in LΨ, and he claim holds. l
Finally, le us no ice ha in case Ψ “A, i.e., he se o e en s we a e conside ing coincides wi h he
domain o he fini e Boolean algeb a we dealing wi h, hen PA, NA, BAand LAa e all polyhed a ha
espec i ely co espond o he se s o p obabili ies, necessi y measu e, belie unc ions and lowe p obabili ies
on A.
3.3. A cohe ence-wise compa ison
We now compa e he a ious defini ions o cohe ence a ising om he Du ch Book me hod ecalled in
Sec ion 2. Mo e p ecisely we will ca y ou a compa ison o he co esponding ex endabili y heo ems.
A fi s esul in his di ec ion, whe e books a e conside ed only on he whole algeb a 4o wo a oms, is
Co olla y 3.2. This es ablishes ha on 4e e y lowe p obabili y is a belie unc ion.
18 E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
Fig. 3. (le -mos ) The poly ope PΨ; (cen e ) The opical poly ope NΨ; ( igh -mos ) The poly opes BΨand LΨ.
No e ha his does no apply in gene al o o he unce ain y measu es. I is indeed easy o see ha on
4, p obabili ies, necessi y measu es and belie unc ions can be dis inguished. In o he wo ds, and adop ing
he no a ion in oduced in Subsec ion 3.2, P4XN4‰P4, P4XN4‰N4(meaning ha on 4 he e
a e p obabili ies ha a e no necessi y measu es and ice- e sa) and P4, N4ĂB4(i.e., he e a e belie
unc ions on 4 ha nei he a e p obabili ies, no necessi y measu es). By a ca dinali y a gumen , he same
ela ions hold among PA, NAand BA o all Boolean algeb a Awi h mo e han 4 elemen s. Fo la e use,
le us hence s a e he ollowing esul ha complemen s Co olla y 3.2.
P oposi ion 3.7. Fo e e y Boolean algeb a Awi h ca dinali y |A| ą4, PAXNA‰PA, PAXNA‰NA
and PA, NAĂBAĂLA.
I , ins ead o ull measu es on an algeb a, we conside books on se s o e en s, a fi s i ial, ye sugges i e,
obse a ion is he ollowing: i Ais any Boolean algeb a and Ψ “ au, hen PΨ“NΨ“BΨ“LΨ. The
same esul applies i ially o se s o e en s o he o m Ψ “ K, a, Ju o Ψ “ K, ausince all unce ain y
measu es conside ed so a a e no malized, whence hey all assign 0 o Kand 1 o J. Fo his eason i
will be impo an o define when a se o e en s is adequa e o he analysis we p opose. Those se s will be
o mally defined below.
In ligh o he abo e easy ema k, we now aim a in es iga ing he obus ness o ex ension heo ems
o unce ain y measu es and, by doing so, a unde s anding up o which ex en he abo e P oposi ion 3.7
gene alizes o adequa e se s o e en s.
The nex example has inspi ed he esul desc ibed by ou main esul , namely Theo em 3.10.
Example 3.8. Le Abe he Boolean algeb a o 8 elemen s and 3 a oms α1, α2, α3uand conside he non-
i ial se o e en s Ψ “ a1, a2, a3u ĂAwhe e a1“α1_α2, a2“α2_α3and a3“α1_α3. The algeb a
Ahas 3 homomo phisms o 0, 1u. Compu ing he poin s e1, e2, e3as in (14), we ge
e1“p1,0,1q;e2“p1,1,0q;e3“p0,1,1q.
The polyhed a PΨ, NΨ, BΨand LΨa e hence as in Fig. 3.
No ice ha , al hough Ψdoes no coincide wi h he whole algeb a A, i allows o dis inguish hose
books ha a e ei he ex endible o a p obabili y o a no malized necessi y, om hose ex endible o belie
unc ions o lowe p obabili ies. Indeed bo h PΨand NΨa e s ic subse s o BΨand LΨ. In e es ingly,
in his specific example, BΨand LΨcoincide.
We now define he no ion o “adequa e” se o e en s Ψwhich allows us o disca d hose cases ha we
al eady know do no allow us o dis inguish BΨ om LΨ.
Defini ion 3.9. Le Abe a Boolean algeb a. Anon-emp y subse Ψo Ais adequa e i Ψis a s ic subse
o Az K, Ju and he subalgeb a AΨo Agene a ed by Ψhas a leas h ee a oms.
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 19
Ou main esul shows ha pa ial assignmen s on e en s exis o which i is impossible o ell whe he
hey a e cohe en in he sense o lowe p obabili y heo y bu ail cohe ence acco ding o belie unc ions.
In logical e ms, his sugges s ha he e a e non-negligible limi s o he exp essi e powe o cohe ence.
In o he wo ds, we can show ha o e e y algeb a Awi h a leas h ee a oms he e exis s Ψ ĂAs. .
he con ex hull cha ac e ising he assignmen s βon Ψ ex endible o p obabili y measu es o e Ais no
included in he con ex hull cha ac e ising he assignmen s ex endible o necessi y measu es o e Aand ice
e sa. I.e., he e a e β1, β2:Ψ Ñ 0, 1ss. . β1PPΨbu β1RNΨ, and β2PNΨbu β2RPΨ. In he
same se ing, we could expec a simila beha iou also o he mo e gene al unce ain y measu es o belie
unc ions and lowe p obabili ies. Howe e , as Theo em 3.10 shows, his is no he case. In ac , we will
show ha he con ex hull cha ac e ising he assignmen s on Ψ ha a e b -cohe en (BΨ) coincides wi h
he con ex hull cha ac e ising he assignmen s on Ψ ha a e l-cohe en (LΨ). Howe e , i βPBΨ“LΨ,
hen he co esponding ex ensions β1PBAand β2PLAmigh no be he same.
Theo em 3.10 (When BΨ“ LΨ). Fo e e y algeb a Awi h a leas h ee a oms he e exis s an adequa e
subse Ψo Asuch ha PΨXNΨ‰PΨand PΨXNΨ‰NΨ, bu BΨ“LΨ.
P oo . Le us assume wi hou loss o gene ali y ha α1, ..., αn(n ě3) a e he a oms o Aand le us fix
he subse Ψo Amade o he ollowing elemen s: a1“α1_α2, a2“α1_α3and a3“α2_α3. Clea ly Ψ
is adequa e in he sense o Defini ion 3.9.
Fi s , le us show ha PΨXNΨ‰PΨand PΨXNΨ‰NΨ.
By P oposi ion 3.6, BΨĎLΨ. Thus, le βbe a book in LΨ. We wan o p o e ha βPBΨ. Le Pbe
a lowe p obabili y on Asuch ha , o all i “1, ..., 3, Ppaiq “βpaiq. Le us also assume ha Pis no a
p obabili y, ha is o say, ha βdoes no belong o PΨ, o he wise, he claim would be i ial.
Now we p o e he ollowing.
Fac 3.11. βPM“copmin e1, e2u, min e2, e3u, min e1, e3u, min e1, e2, e3uq.
P oo o Fac 3.11.Assume, by way o con adic ion, ha βRM. Thus, βP 0, 1s3zM, ha is o say,
βPcope1, e2, e3, max e1, e2, e3uq. In o he wo ds, he e exis λ1, λ2, λ3, λ4(wi h λ4ą0) such ha
řiλi“1
and
β“λ1e1`λ2e2`λ3e3`λ4max e1,e2,e3u.
The exp ession abo e equals λ1e1`λ2e2`λ3e3`max λ4e1, λ4e2, λ4e3uand since a `max b, cu “max a `
b, a `cu, one has
β“max β1,β
2,β
3u
whe e β1“pλ1`λ4qe1`λ2e2`λ3e3, β2“λ1e1`pλ2`λ4qe2`λ3e3, β3“λ1e1`λ2e2`pλ3`λ4qe3.
Thus, β1, β2, β3PPΨ. Le ing Pi, o i “1, 2, 3such ha Piex ends βi, we conclude ha βex ends o an
uppe p obabili y. The e o e, by assump ion βex ends o a lowe p obabili y. In addi ion, βex ends o an
uppe p obabili y, hus βex ends o a p obabili y ha is absu d by a p e ious hypo hesis. l
Now, we go back o he p oo o he main claim and we p o e ha min e1, e2u, min e2, e3u, min e1, e3u,
min e1, e2, e3u PNΨ. The claim is indeed easy o show by di ec compu a ion. Fo ins ance, check ha
min e1, e2, e3u “p0, 0, 0qis pNpa1q, Npa2q, Npa3qq whe e Nis he necessi y measu e compu ed as in (4)
and gi en by he no malized possibili y dis ibu ion π:α “1 o all “1, ..., n.
The e o e βis a con ex combina ion o poin s belonging o NΨ. Hence i ex ends o a belie unc ion,
concluding he p oo o Theo em 3.10.l
20 E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
No ice ha he abo e esul does no say ha i βex ends o a lowe p obabili y P, hen Pis a belie
unc ion. All i shows is ha i βon e en s a1, a2, a3ex ends o lowe p obabili y P, hen he e exis s a
belie unc ion Bel ha ag ees wi h Pon he ai’s bu need no ag ee elsewhe e. The ollowing example
cla ifies his.
Example 3.12. Take again he se up o Example 2.16, now wi h he se o e en s Ψ “ a1, a2, a3udefined as
in he p oo o Theo em 3.10: a1“α1_α2, a2“α2_α3, and a3“α1_α3. Le us conside also he book
β:aiÞÑ q o e e y i “1, 2, 3.
Since qď1{2, qď1 ´qand hence Ppa1q “Ppa2q “Ppa3q “q. Thus, he lowe p obabili y Pdefined
as in (10) ex ends β.
Fu he mo e, Pis no a belie unc ion. Indeed, Ppa1q `Ppa2q `Ppa3q ´Ppa1^a2q ´Ppa2^a3q ´
Ppa1^a3q `Ppa1^a2^a3q. Now, a1^a2^a3“K, whence Ppa1^a2^a3q “0 and, by defini ion o
ai, Ppa1^a2q “Ppa2^a3q “Ppa1^a3q “0. The e o e, since qą1{3, he abo e exp ession educes o
Ppa1q `Ppa2q `Ppa3q “3qą1 “Ppa1_a2_a3qshowing ha Pdoes no sa is y (13).
Howe e , he belie unc ion Bel whose mass assignmen s is mp α1uq “mp α2uq “mp α3uq “q{2,
mp α1, ..., α uq “1 ´3
2qand mpXq “0o he wise, ex ends he same book β o he Boolean algeb a A.
Le us conside he se o e en s Ψ1“ a1, a2, a3, a4, a5, a6uwhe e a1“α1, a2“α2, a3“α3and a4, a5,
and a6a e defined as in he p oo o Theo em 3.10: a4“α1_α2, a5“α2_α3, and a6“α1_α3. The
ec o s e1, e2, and e3 ela i e o Ψ1a e defined as ollows:
e1“p1,0,0,1,0,1q;e2“p0,1,0,1,1,0q;e3“p0,0,1,0,1,1q.
Thus, he e ices o BΨ1a e e1, e2, e3and min e1, e2u “p0, 0, 0, 1, 0, 0q, min e2, e3u “p0, 0, 0, 0, 1, 0q,
min e1, e3u “p0, 0, 0, 0, 0, 1qand min e1, e2, e3u “p0, 0, 0, 0, 0, 0q. We can hen e i y ha he poin
p0, 0, 0, q, q, qq RBΨ1while i belongs o LΨ1.
Rema k 3.13. As we poin ed ou in Sec ion 3.2, an exhaus i e desc ip ion o he poly ope LΨis no known
ye . Mo e p ecisely, al hough LΨis gene a ed by i s ex emal poin s by K ein-Milman Theo em [19, The-
o em 1.2], hose la e , ha a e ex emal lowe p obabili ies cohe en on Ψ, a e no known in gene al.
Howe e , i Ψis a se o e en s o which BΨ“LΨ, he ex emal lowe p obabili ies ha a e cohe en on
Ψcoincide wi h he ex emal belie unc ions ha a e cohe en on he same e en s. In o he wo ds, unde
ha hypo hesis and om he esul s ecalled in Sec ion 3.2, ex emal lowe p obabili ies ha a e cohe en
on Ψa e cohe en necessi y measu es on Ψ, i.e., ex pLΨq ĎNΨ.
4. A me alogical ep esen a ion
In his final sec ion, we will pu o wa d a amewo k o ep esen in logical e ms he no ions o cohe ence
and, mo e in gene al, he unce ain y heo ies we conside ed so a . Mo e in de ails we will see how he
geome ic app oaches de eloped in he p e ious sec ions allow o b idge cohe ence and unce ain y heo ies
on one side, and p oposi ional logic and deduc i e easoning on he o he . Amajo ole, in his sense, will
be played by p oposi ional Riesz logic [16] ha we will b iefly ecall in he nex Subsec ion 4.1, while in
Subsec ion 4.2 we will p esen he connec ion be ween such o malism and cohe ence.
4.1. Riesz consequence ela ion (R
A Riesz space is a ec o space u he endowed wi h a la ice o de ď ha is compa ible wi h he
ec o spaces ope a ions, i.e., a ec o la ice. Riesz p oposi ional logic R ha we will b iefly p esen in his
sec ion, has been fi s ly in oduced in [16]as he ex ension o Łukasiewicz logic [41]by a uncoun able amily
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 21
o una y connec i es ∇ ( o P 0, 1s) axioma ized in such a way ha , om he algeb aic iewpoin , hey
a e necessa ily in e p e ed as he scala p oduc in a(n in e al o a) ec o la ice.
Gi ing an exhaus i e desc ip ion o he logic Ris ou o he scope o he p esen pape , and we in i e
he in e es ed eade o consul he li e a u e on his subjec (c . [16]). Howe e , wha is o key impo ance
o ou logical analysis o cohe ence, is o ecall in wha Ris able o speak and eason abou polyhed al
geome y (see [15]and [14, §3.3]). On his la e aspec , we will ocus he es o his subsec ion. In o de
o imp o e eadabili y, we will assume he eade o be amilia wi h he basic o Łukasiewicz logic and
MV-algeb as. Fo a mo e exhaus i e in oduc ion abou his subjec we in i e he eade o consul [5].
Conside he eal uni in e al 0, 1sand he algeb aic language p‘, , 0qo ype p2, 1, 0qand whe e, o all
x, yP 0, 1s, x ‘y“min 1, x `yuand x “1 ´x. Fu he mo e, o e e y P 0, 1sle p : 0, 1s Ñ 0, 1sbe
he una y ope a ion x ÞÑ p pxq “ x. The algeb a 0, 1sRMV “p 0, 1s, ‘, , p u P 0,1s, 0qis he p o o ypical
example o a Riesz MV-algeb a and i is called he s anda d Riesz MV-algeb a.
Defini ion 4.1. A sys em Aon a non-emp y domain Aand in he algeb aic language p‘, , p u P 0,1s, 0qis
a Riesz MV-algeb a i Abelongs o Vp 0, 1sRMV q “RMV , he algeb aic a ie y gene a ed by 0, 1sRMV .
Fo he logical ansla ion o cohe ence ha we will p esen in his final sec ion, amajo ole will be played
by special RMV-algeb as: he fini ely gene a ed ee algeb as. These s uc u es a e, up o isomo phism,
he Lindenbaum-Ta ski algeb a o Riesz logic ha we will b iefly p esen below, and hey can be u he
cha ac e ized as ollows.
Example 4.2 ([15, Theo em 1.3]). Le kbe fini e. A unc ion : 0, 1skÑ 0, 1sis said o be a Riesz unc ion
i is con inuous, piecewise linea and each piece has coefficien s om R. Then, o e e y fini e kPN, le
Rpkq he se o all Riesz unc ions on 0, 1sk. The k-gene a ed ee Riesz MV-algeb a is, up o isomo phism,
he algeb a Rpkq “pRpkq, ‘, , p u P 0,1s, 0qwhe e ope a ions a e defined by he poin -wise applica ion o
hose o he s anda d algeb a 0, 1sRMV .
Riesz logic $Ris he algeb aizable logic, in he sense o Blok and Pigozzi [4], whose equi alen algeb aic
seman ics is RMV . The algeb aizabili y o Rw. . . RMV gi es us ha o mulas o he o me can be
equi alen ly ega ded as e ms o he la e . In wha ollows we will hence say ha ˆϕ(possibly ˆϕpx1, ..., xkq
i we wan o poin ou he p oposi ional a iables occu ing in i )1is a o mula o R, meaning ha ˆϕ
is a e m in he algeb aic language o Riesz-algeb a (on a iables x1, ..., xk). As we ecalled abo e, he
Lindenbaum-Ta ski algeb a o Ron kp oposi ional a iable Lpkqcoincides, up o isomo phism, wi h he
k-gene a ed ee Riesz MV-algeb a Rpkqo Example 4.2.
The isomo phism be ween Lpkqand Rpkq, ells us ha i ˆϕpx1, ..., xkqis a o mula o Riesz logic, i s
equi alence class (modulo equi-p o abili y in R) in Lpkqcan be ega ded as a Riesz unc ion ˆϕ: 0, 1skÑ
0, 1sand, ice e sa, o e e y Riesz unc ion PRpkq, he e exis s a (possibly no unique) o mula
ˆϕ px1, ..., xkqwhose equi alence class in Lpkqcan be isomo phically associa ed o .
Defini ion 4.3. Fo e e y fini e kand e e y o mula ˆϕpx1, ..., xkq, we will w i e Modpˆϕq “ pa1, ..., akq P
0, 1sk| ˆϕpa1, ..., akq “1u. This se will be called he se o models o ˆϕo , equi alen ly, he one-se o ˆϕ.
The seman ic consequence ela ion o Rwill be deno ed by (R. Hence, i ˆϕpx1, ..., xkqand
ˆ
ψpx1, ..., xkq
a e o mulas, ˆϕ(Rˆ
ψmeans ha Modpˆϕq ĎModpˆ
ψq. In o he wo ds, ˆ
ψpa1, ..., akq “1in 0, 1sRMV o
all hose pa1, ..., akq P 0, 1sksuch ha ˆϕpa1, ..., akq “1in 0, 1sRMV ; ˆϕ)(Rˆ
ψs ands o ˆϕ(Rˆ
ψand
ˆ
ψ(Rˆϕand i hence indica es ha Modpˆϕq “Modpˆ
ψq.
1In his las sec ion we will use he no a ion ˆϕ,
ˆ
ψe c., o dis inguish o mulas o R om he elemen s o an a bi a y Boolean
algeb as ha , as in he p e ious sec ions, we will deno e by lowe case G eek le e s ϕ, ψ, e c.

22 E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
The nex esul summa ises known and use ul ac s abou Riesz logic and i s close connec ion wi h
polyhed al geome y.
P oposi ion 4.4 ([15, Theo em 3.3]). (1) Fo e e y o mula ˆϕpx1, ..., xkq, Modpˆϕqis a polyhed on o 0, 1sk;
(2) Fo e e y polyhed on Po 0, 1sk he e exis s a o mula ˆρPpx1, ..., xkqsuch ha P“ModpˆρPq.
Recall ha he o mula ˆρPo he abo e P oposi ion 4.4 is no necessa ily unique. Howe e , in wha
ollows, we will speak abou he o mula ˆρPsuch ha P“ModpˆρPqin ending ha we ha e chosen one
among all hose o mulas ha sa is y he abo e claim.
The las use ul defini ion ha is necessa y o ecall is ha o R- unc ion be ween polyhed a.
Defini ion 4.5. Le PĎ 0, 1skand QĎ 0, 1snbe polyhed a. We say ha a map η:PÑQis a R-map i
he e exis s Riesz unc ions 1, ..., nPRpkqsuch ha , o all x PP,
ηpxq“p 1pxq,...,
npxqq P Q.
Fo wha we will show in he nex subsec ion is impo an o no ice ha he p ojec ion maps o any
polyhed on PĎ 0, 1sk o a lowe dimension 0, 1sn( o n ďk) a e elemen a y examples o R-maps.
4.2. Cohe ence h ough (R
The esul s p esen ed in Subsec ion 3.2, and Theo em 3.4 in pa icula , show ha o e e y fini e se o
e en s Ψ “ ψ1, ..., ψku, he se s PΨ, NΨ, BΨand LΨo books ha ex ends o p obabili y unc ions,
necessi y measu es, belie unc ions and lowe p obabili ies espec i ely, a e polyhed a o 0, 1sk. The ol-
lowing esul p o ides p elimina y esul s on he logical desc ip ion o cohe ence ia Riesz logic R. I s p oo
is an immedia e consequence o P oposi ion 4.4 and he defini ion o (R. In wha ollows we will adop he
no a ion used in P oposi ion 4.4 (2).
Co olla y 4.6. Fo e e y fini e se o e en s Ψ “ a1, ..., akuand o e e y book β:Ψ Ñ 0, 1s, he ollowing
condi ions hold:
1. pβpa1q, ..., βpakqq PCΨiff ˆρ βu(RˆρCΨand o e e y CP P, N, B, Lu;
2. ˆρPΨ(RˆρBΨ; ˆρNΨ(RˆρBΨ; ˆρBΨ(RˆρLΨ.
No ice ha , while claim (1) in he co olla y abo e ollows om he ac ha poin s a e special examples
o polyhed a, claim (2) is a consequence o wha we obse ed in Sec ion 3. P ecisely ha cohe ence o lowe
p obabili ies is a mo e gene al no ion han cohe ence o belie unc ions and his la e , in u n, is mo e
gene al han cohe ence o necessi y measu es and p obabili y unc ions.
Also he main esul o Sec ion 3on he exis ence o adequa e se s o e en s o which one canno
dis inguish be ween books ha a e ex endible o belie unc ions o lowe p obabili ies, Theo em 3.10, can
be eph ased in he ollowing e ms.
P oposi ion 4.7. Fo e e y Boolean algeb a Awi h a leas h ee a oms he e exis s an adequa e subse Ψo
Asuch ha :
1. ˆρPΨ*ˆρPΨXNψand ˆρNΨ*ˆρPΨXNψ;
2. ˆρBΨ)(RˆρLΨ.
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 23
So a , we ha e seen how he geome ic desc ip ion o cohe ence can be s aigh o wa dly desc ibed by
me alogical p ope ies o Riesz logic. Now, we end his sec ion by showing a less i ial in e p e a ion ha
allows o ega d polyhed a o cohe en books as p ojec ions o ull measu es defined on fini e algeb as.
In wha ollows, le us use he symbol C o be any among P, N, B, Luand, once we ha e fixed a
CP P, N, B, Luwe will say ha a map d om a Boolean algeb a A o 0, 1sis a C-measu e as a gene al
nomencla u e o : dis a fini ely addi i e p obabili y measu e (in case C“P), dis a necessi y measu e (i
C“N), dis a belie unc ion ( o C“B), and dis a lowe p obabili y (i C“L).
F om he nex esul , whose p oo is a di ec consequence o he ex ension Theo em 3.4 and he defini ion
o LΨ, we will s a deno ing by πΨ he p ojec ion map o 0, 1sA o 0, 1sΨ.
P oposi ion 4.8. Fo e e y fini e Boolean algeb a Aand e e y subse Ψo A,
CΨ“πΨpCAq
The e o e, each poly ope CΨis he p ojec ion o CAon he axes indexed by a1, ..., ak.
F om wha we obse ed a he end o Subsec ion 4.1 p ojec ions o polyhed a o lowe dimensional cubes
a e R-maps. The e o e, by Defini ion 4.5, o e e y Ψ ĎAand o e e y CΨĎ 0, 1sk, he e exis Riesz
unc ions 1, ..., k: 0, 1sAÑ 0, 1ssuch ha he p ojec ion map πΨ:CAĎ 0, 1s2nÑCΨĎ 0, 1skac s
as ollows: o e e y pa1, ..., a2nq PCA,
πΨpa1,...,a
2nq“p 1pa1,...,a
2nq,...,
kpa1,...,a
2nqq.
Thus, p 1pa1, ..., a2nq, ..., kpa1, ..., a2nqq PCΨ. Mo e de ails on wha such p ojec ions look like om he
logico-algeb aic pe spec i e will be gi en in he p oo o he nex esul ha p o ides a mo e p ecise logical
eading o he p e ious P oposi ion 4.8.
P oposi ion 4.9. Le Abe he fini e Boolean algeb a o ca dinali y 2nand le Ψ “ a1, ..., aku ĎA. Then,
he e a e Riesz unc ions 1, ..., k: 0, 1sAÑ 0, 1ssuch ha pb1, ..., bkq PCΨi and only i he e exis s
pa1, ..., a2nq PCAsuch ha
pb1,...,b
kq“p 1pa1,...,a
2nq,...,
kpa1,...,a
2nqq.
P oo . Le us fix, wi hou loss o gene ali y, he enume a ion a1, ..., a2n o he elemen s o Ain such a
way ha he fi s kelemen , in he na u al o de , de e mine he e en se Ψ (clea ly kď2nsince Ψ ĎA).
Fo e e y i “1, ..., k, le ibe he map om 0, 1sA o 0, 1sdefined as ollows: o all g:A Ñ 0, 1s,
ipgq “gpaiq. In o he wo ds, once iden ified he unc ions om A o 0, 1sas s ings o leng h 2no elemen s
o 0, 1sas pa1, ..., a2nq, ipa1, ..., a2nq “ai. Thus, iis a Riesz homomo phism o 0, 1sA o 0, 1s.
Now, i pb1, ..., bkq PCΨ, by defini ion he e exis s a C-measu e d :A Ñ 0, 1s ha ex ends i . Tha is
o say, pb1, ..., bkq PCΨi and only i he ec o pdpa1q, dpa2q, ..., dpa2nqq PCAand o all i “1, ..., k,
bi“dpaiq. The e o e, o all i “1, ..., k, by defini ion o i, one has
bi“dpaiq“ ipdpa1q,...,dpakqq.
Con e sely, i pb1, ..., b2nq PCA, hen he e exis s a C-measu e d :A Ñ 0, 1ssuch ha , o all j“1, ..., 2n,
bj“dpajq. By defini ion o i, and by ou p e ious assump ion on he ai’s o i “1, ..., k,
p 1pa1,...,a
2nq,...,
kpa1,...,a
2nqq “ pb1,...,b
kq“pdpa1q,...,dpakqq.
The la e is cohe en being he es ic ion o Ψo a C-measu e d. Thus we finally ge p 1pa1, ..., a2nq, ...,
kpa1, ..., a2nqq PCΨand he claim is se led. l
24 E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342
Ou nex esul p o ides a uni o m logical ep esen a ion o he ex ension heo ems o p obabili ies,
necessi y measu es, belie unc ions and lowe p obabili ies. Mo e p ecisely, he nex con ains wo esul s: he
fi s one desc ibes, in logical e ms, he claim o he abo e P oposi ions 4.8 and 4.9 by in e p e ing geome ic
p ojec ions as logical subs i u ions2; he second one b idges he claim o Theo em 3.4 ha cha ac e izes
ex ension esul s ia he geome y o cohe ence and ha o Co olla y 4.6 ha links he geome y o cohe ence
and deduc ions in Riesz logic. Mo eo e , i is wo h no icing ha he cha ac e iza ion we p esen in he
nex heo em ex ends [21, Theo em 5.5] in he scope o unce ain y heo ies ha allow o such logical
desc ip ion.
Theo em 4.10. Le Abe a fini e Boolean algeb a and Ψ ĎA. Then he e exis s a subs i u ion σΨsuch ha
(1) σΨpˆρCΨq )(RˆρCA.
Fu he mo e, o all β:Ψ Ñ 0, 1s he ollowing condi ions a e equi alen :
(2) βex ends o a C-measu e;
(3) ˆρ βu(RˆρCΨ;
P oo . Le us no ice ha he equi alence be ween (2) and (3) immedia ely ollows om Theo em 3.4 and
Co olla y 4.6 (1).
As o p o e (1), by P oposi ion 4.9, he e a e Riesz unc ions 1, ..., k(whe e k“|Ψ|) such ha ,
CΨ“ p 1pa1,...,a
2nq,...,
kpa1,...,a
nqq | pa1,...,a
2nqPCAu,
whe e 2n“|A|.
The e o e, i x1, ..., xka e he a iable occu ing in ˆρCΨ, define σΨ o be he subs i u ion ha maps
each a iable xi o he e m ˆϕ ipy1, ..., y2nq, he Riesz o mula ha co esponds o i. Mo e p ecisely, σΨ
maps ˆρCΨpx1, ..., xkq o
σΨpˆρCΨqpy1,...,y
2nq“ˆρCΨpˆϕ 1py1,...,y
2nq,..., ˆϕ kpy1,...,y
2nqq
Then he claim ollows. Indeed, by P oposi ion 4.9 and P oposi ion 4.4(2), one has:
pa1,...,a
2nqPModpˆρCAqiff pa1,...,a
2nqPCA
iff p 1pa1,...,a
2nq,...,
kpa1,...,a
nqq P CΨ
iff p 1pa1,...,a
2nq,...,
kpa1,...,a
nqq P ModpˆρCΨq
iff pa1,...,a
2nqPModpˆρCΨpˆϕ 1,..., ˆϕ kq
iff pa1,...,a
2nqPModpσΨpˆρCΨq.
The e o e, pa1, ..., a2nq PModpˆρCAqiff pa1, ..., a2nq PModpσΨpˆρCΨqq. In o he wo ds, ModpˆρCAq “
ModpσΨpˆρCΨqq and hence, by defini ion o )(R, i ollows ha σΨpˆρCΨq )(RˆρCA.l
5. Conclusions and u u e wo k
We ha e pu o wa d a logico-geome ic amewo k which allows us o in es iga e he no ion o cohe ence
a a conside able le el o gene ali y and de ail. Wi hin his amewo k we ha e pu o wa d i) a compa ison
2Recall ha in he amewo k o algeb aic logic, gi en a language L, alogical subs i u ion (o simply a subs i u ion) is a
map σ ha assigns o e e y p oposi ional a iable xo L, a o mula ˆϕo he same language L. Equi alen ly, subs i u ions a e
endomo phisms o he Lindenbaum-Ta ski algeb a o L, once es ic ed o he a iables o L.
E.A. Co si e al. / Annals o Pu e and Applied Logic 175 (2024) 103342 25
be ween he geome ic ep esen a ions o cohe ence o fini ely-addi i e measu es wi h hei non-addi i e
coun e pa s, and ii) a compa ison be ween non-addi i e measu es hemsel es. Ou key finding is ha
non-addi i e measu es which can be dis inguished axioma ically may no be dis inguishable cohe ence-wise.
The ou comes o ou geome ical compa isons a e also eco e ed wi hin a logical ep esen a ion o cohe ence
p o ided by means o sui ably defined Riesz consequence ela ions.
Two ques ions which we hink a e wo hy o u he in es iga ion a ise in he amewo k pu o wa d in
his pape .
In Chap e 3 o [12], B uno de Fine i es ablishes he equi alence be ween he Du ch Book me hod
discussed abo e and he me hod o (p ope ) sco ing ules. This la e defines cohe ence as he minimisa ion
o expec ed loss unde a well-defined penal y unc ion known as he B ie Sco e. As de Fine i poin s ou ,
he equi alence be ween he wo seemingly diffe en c i e ia has a geome ic explana ion, as hey bo h boil
down o pinning he con ex hull o he n-dimensional linea space Swhich a ises om assigning alues
in 0, 1s o a se o ne en s Ψ. A na u al ques ion hen is o ex end he cohe ence-wise compa ison o
non-addi i e unce ain y measu es ca ied ou in his pape o he sco ing ules me hod. Ou p elimina y
in es iga ions on his show ha he answe is no s aigh o wa d. In [8], we pu o wa d a sco ing ule o
belie unc ions and show ha a cha ac e isa ion o cohe ence as minimisa ion o expec ed loss unde ha
ule in he con ex o opical geome y. To ca y ou he analogue o he p esen compa ison a sui able
sco ing ule o lowe p obabili ies mus be pu o wa d. Some ea lie esul s o Seiden eld Sche ish and
Kadane [48]poin ou se e al difficul ies in doing his. In addi ion, wi h he excep ion o wha we ha e
no ed in Rema k 3.13, a ull desc ip ion o he ex eme poin s o LΨis no p esen ly known. Hence mo e
esea ch in his di ec ion is needed o pu o wa d an analysis simila o he p esen one in e ms o sco ing
ules.
The me alogical amewo k o Sec ion 4allows us o pu sue an analogy which has been used casually
in his pape . The idea is o ake he p ope ies o he be ing game as fixing he “in ended seman ics” o
an unce ain y ep esen a ion. To un old he analogy, ecall ha in classical logic, model inclusion is he
in ended seman ics o he classical consequence ela ion “(”. Simila ly, p o abili y is he in ended seman ics
o i s in ui ionis ic coun e pa , and so on. The ole o in ended seman ics is chiefly o p o ide guidelines
o he ma e ial adequacy o he o mal defini ion, o bo ow Ta ski’s own exp ession. So in he case o de
Fine i’s Du ch Book a gumen , his jus ifies “incu ing su e loss” as a bla an ly undesi able ou come o a
bookmake . Hence cohe ence is defined in such a way o a oid ha . This sugges s he ollowing ques ion: a e
he e basic p ope ies ha can be iden ified as c ucial in mo ing om one unce ain y measu e o ano he ,
p e y much he way in which he amily o no mal modal logics a ise om adding sui able condi ions o he
dis ibu ion axiom K? In o he wo ds, can we use he logical amewo k in oduced abo e o pu o wa d
a modula app oach enabling us o ma ch, whene e possible, dis inc no ions o cohe ence wi h dis inc
p ope ies o he unde lying decision p oblem? We know om ou main esul ha his is no going o ma ch
ully, and gene ally, he axioma ic-wise dis inc measu es o unce ain y, bu no hing in p inciple p e en s
he emaining cases o be desc ibed me alogically along he lines o Theo em 4.10.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha
could ha e appea ed o influence he wo k epo ed in his pape .
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.